1.introduction. - tu berlin

SINGLE MACHINE SCHEDULING WITH RELEASE DATES∗

MICHEL X. GOEMANS† , MAURICE QUEYRANNE‡ , ANDREAS S. SCHULZ§ ,MARTIN SKUTELLA¶, AND YAOGUANG WANG‖

SIAM J. DISCRETE MATH. c© 2002 Society for Industrial and Applied MathematicsVol. 15, No. 2, pp. 165–192

Abstract. We consider the scheduling problem of minimizing the average weighted completiontime of n jobs with release dates on a single machine. We first study two linear programmingrelaxations of the problem, one based on a time-indexed formulation, the other on a completion-time formulation. We show their equivalence by proving that a O(n logn) greedy algorithm leads tooptimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, aconcept which enhances our understanding of these LP relaxations. Based on the greedy solution, wedescribe two simple randomized approximation algorithms, which are guaranteed to deliver feasibleschedules with expected objective function value within factors of 1.7451 and 1.6853, respectively, ofthe optimum. They are based on the concept of common and independent α-points, respectively. Theanalysis implies in particular that the worst-case relative error of the LP relaxations is at most 1.6853,and we provide instances showing that it is at least e/(e − 1) ≈ 1.5819. Both algorithms may bederandomized; their deterministic versions run in O(n2) time. The randomized algorithms also applyto the on-line setting, in which jobs arrive dynamically over time and one must decide which job toprocess without knowledge of jobs that will be released afterwards.

Key words. approximation algorithm, LP relaxation, scheduling, on-line algorithm

AMS subject classifications. 90C27, 68Q25, 90B35, 68M20

PII. S089548019936223X

1. Introduction. We study the single-machine scheduling problem with releasedates in which the objective is to minimize a weighted sum of completion times. It isdefined as follows. A set N = {1, 2, . . . , n} of n jobs has to be scheduled on a singledisjunctive machine. Job j has a processing time pj > 0 and is released at time rj ≥ 0.We assume that release dates and processing times are integral. The completion timeof job j in a schedule is denoted by Cj . The goal is to find a nonpreemptive schedulethat minimizes

∑j∈N wjCj , where the wj ’s are given positive weights. In the classical

scheduling notation [12], this problem is denoted by 1| rj |∑

wjCj . It is strongly NP-hard, even if wj = 1 for all jobs j [17].

∗Received by the editors October 1, 1999; accepted for publication (in revised form) November15, 2001; published electronically February 27, 2002.

http://www.siam.org/journals/sidma/15-2/36223.html†Department of Mathematics, Room 2-351, MIT, 77 Massachusetts Avenue, Cambridge, MA

02139 ([email protected]). The research of this author was performed partly when the authorwas at C.O.R.E., Louvain-la-Neuve, Belgium and was supported in part by NSF contract 9623859-CCR.

‡Faculty of Commerce and Business Administration, University of British Columbia, Vancouver,BC, Canada V6T 1Z2 ([email protected]). The research of this author wassupported in part by a research grant from the NSERC (Natural Sciences and Research Council ofCanada) and by the UNI.TU.RIM. S.p.a. (Societa per l’Universita nel riminese).

§Sloan School of Management, Room E53-361, MIT, 77 Massachusetts Avenue, Cambridge, MA02139 ([email protected]). The research of this author was performed partly when he was with theDepartment of Mathematics, Technische Universitat Berlin, Germany.

¶Fakultat II—Mathematik und Naturwissenschaften, Institut fur Mathematik, MA 6-1, Technis-che Universitat Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany ([email protected]).This author was supported in part by DONET within the frame of the TMR Programme (contractERB FMRX-CT98-0202) while staying at C.O.R.E., Louvain-la-Neuve, Belgium, for the academicyear 1998/99.

‖Peoplesoft, Inc., Pleasanton, CA 94588 (Yaoguang [email protected]). This author wassupported by a research fellowship from Max-Planck Institute for Computer Science, Saarbrucken,Germany.

165

166 GOEMANS, QUEYRANNE, SCHULZ, SKUTELLA, AND WANG

One of the key ingredients in the design and analysis of approximation algorithmsas well as in the design of implicit enumeration methods is the choice of a bound onthe optimal value. Several linear programming-based as well as combinatorial lowerbounds have been proposed for this well-studied scheduling problem; see, for example,Dyer and Wolsey [9], Queyranne [22], and Queyranne and Schulz [23], as well asBelouadah, Posner, and Potts [4]. The LP relaxations involve a variety of differenttypes of variables which, e. g., express whether either job j is completed at time t(nonpreemptive time-indexed relaxation), or whether it is being processed at time t(preemptive time-indexed relaxation), or when job j is completed (completion timerelaxation). Dyer and Wolsey show that the nonpreemptive time-indexed relaxationis stronger than the preemptive time-indexed relaxation. We will show that the latterrelaxation is equivalent to the completion time relaxation that makes use of the so-called shifted parallel inequalities. In fact, it turns out that the polyhedron definedby these inequalities is supermodular, and hence one can optimize over it by usingthe greedy algorithm. A very similar situation arises in [24]. The greedy solution mayactually be interpreted in terms of the following preemptive schedule, which we callthe LP schedule: at any point in time it schedules among the available jobs one withthe largest ratio of weight to processing time. Uma and Wein [38] point out that thevalue of this LP solution coincides with one of the combinatorial bounds of Belouadah,Posner, and Potts based on the idea of allowing jobs to be split into smaller piecesthat can be scheduled individually.

We show that the optimal value of 1| rj |∑

wjCj is at most 1.6853 times thelower bound given by any of these three equivalent relaxations—the preemptive time-indexed relaxation, the completion time relaxation, or the combinatorial relaxationin [4]. We prove this result on the quality of these relaxations by converting the(preemptive) LP schedule into a nonpreemptive schedule. This technique leads toapproximation algorithms for 1| rj |

∑wjCj . Recall that a ρ-approximation algorithm

is a polynomial-time algorithm guaranteed to deliver a solution of cost at most ρ timesthe optimal value. A randomized ρ-approximation algorithm is a polynomial-timealgorithm that produces a feasible solution whose expected objective function value iswithin a factor of ρ of the optimal value.

The technique of converting preemptive schedules to nonpreemptive schedulesin the design of approximation algorithms was introduced by Phillips, Stein, andWein [21]. More specifically, for 1| rj |

∑wjCj they showed that list scheduling in

order of the completion times of a given preemptive schedule produces a nonpreemp-tive schedule while increasing the total weighted completion time by at most a factorof 2. In the same paper they also introduced a concept of α-points. This notion wasalso used by Hall, Shmoys, and Wein [14] in connection with the nonpreemptive time-indexed relaxation of Dyer and Wolsey to design approximation algorithms in variousscheduling environments. For our purposes, the α-point of job j in a given preemp-tive schedule is the first point in time at which an α-fraction of j has been completed.When one chooses different values of α, sequencing in order of nondecreasing α-pointsin the same preemptive schedule may lead to different nonpreemptive schedules. Thisincreased flexibility led to improved approximation algorithms: Chekuri et al. [6] for1| rj |

∑Cj and Goemans [11] for 1| rj |

∑wjCj chose α at random and analyzed the

expected performance of the resulting randomized algorithms. We will show that,using a common value of α for all jobs and an appropriate probability distribution,sequencing in order of α-points of the LP schedule has expected performance no worsethan 1.7451 times the optimal preemptive time-indexed LP value. We also prove that

SINGLE MACHINE SCHEDULING WITH RELEASE DATES 167

Table 1Summary of approximation bounds for 1| rj |

∑wjCj . An α-schedule is obtained by sequenc-

ing the jobs in order of nondecreasing α-points of the LP schedule. The use of job-dependent αj ’syields an (αj)-schedule. The results discussed in this paper are below the second double line. Subse-quently, Anderson and Potts [2] gave a deterministic 2-competitive algorithm. For the unit-weightproblem 1| rj |

∑Cj , the first constant-factor approximation algorithm is due to Phillips, Stein, and

Wein [21]. It has performance ratio 2, and it also works on-line. Further deterministic 2-competitivealgorithms were given by Stougie [36] and Hoogeveen and Vestjens [15]. All these algorithms areoptimal for deterministic on-line algorithms [15]. Chekuri et al. [6] gave a randomized e/(e − 1)-approximation algorithm, which is optimal for randomized on-line algorithms [37, 39].

Reference and/or Off-line On-linetype of schedule deterministic randomized deterministic

Phillips et al. [21] 16 + εHall et al. [14] 4 4 + εSchulz [26] 3Hall et al. [13] 3 3 + εChakrabarti et al. [5] 2.8854 + ε 2.8854 + εCombining [5] and [13] 2.4427 + ε 2.4427 + ε

α-schedule

for α = 1/√

2[11] 2.4143 2.4143

[11] 2Best α-schedule

1.7451(random) (αj)-schedule 1.6853 1.6853

by selecting a separate value αj for each job j, one can improve this bound to afactor of 1.6853. Our algorithms are inspired by and partly resemble the algorithmsof Hall, Shmoys, and Wein [14] and Chekuri et al. [6]. In contrast to Hall, Shmoys,and Wein we exploit the preemptive time-indexed LP relaxation, which, on the onehand, provides us with highly structured optimal solutions and, on the other hand,enables us to work with mean busy times. We also use random α-points. The algo-rithm of Chekuri et al. starts from an arbitrary preemptive schedule and makes useof random α-points. They relate the value of the resulting α-schedule to that of thegiven preemptive schedule and not to that of an underlying LP relaxation. Whiletheir approach gives better approximations for 1| rj |

∑Cj and structural insights on

limits of the power of preemption, the link of the LP schedule to the preemptive time-indexed LP relaxation helps us to obtain good approximations for the total weightedcompletion time.

Variants of our algorithms also work on-line when jobs arrive dynamically overtime and, at each point in time, one has to decide which job to process withoutknowledge of jobs that will be released afterwards. Even in this on-line setting, wecompare the value of the computed schedule to the optimal (off-line) schedule andderive the same bounds (competitive ratios) as in the off-line setting. See Table 1 foran account of the evolution of off-line and on-line approximation results for the singlemachine problem under consideration.

The main ingredient to obtain the results presented in this paper is the ex-ploitation of the structure of the LP schedule. Not surprisingly, the LP sched-ule does not solve the strongly NP-hard [16] preemptive version of the problem,1| rj , pmtn |∑wjCj . However, we show that the LP schedule solves optimally thepreemptive problem with the related objective function

∑j wjMj , where Mj is the

mean busy time of job j, i.e., the average point in time at which the machine is busyprocessing j. Observe that, although 1| rj , pmtn |∑wjCj and 1| rj , pmtn |∑wjMj

are equivalent optimization problems in the nonpreemptive case (since Cj = Mj+pj

2 ),


they are not equivalent when considering preemptive schedules.

The approximation techniques presented in this paper have also proved usefulfor more general scheduling problems. For the problem with precedence constraints1| rj , prec |

∑wjCj , sequencing jobs in order of random α-points based on an opti-

mal solution to a time-indexed LP relaxation leads to a 2.7183-approximation algo-rithm [27]. A 2-approximation algorithm for identical parallel machine schedulingP | rj |

∑wjCj is given in [28]; the result is based on a time-indexed LP relaxation, an

optimal solution of which can be interpreted as a preemptive schedule on a fast singlemachine; jobs are then assigned randomly to the machines and sequenced in orderof random αj-points of this preemptive schedule. For the corresponding schedulingproblem on unrelated parallel machines R | rj |

∑wjCj , a performance guarantee of 2

can be obtained by randomized rounding based on a convex quadratic programmingrelaxation [33], which is inspired by time-indexed LP relaxations like the one discussedherein [28]. We refer to [32] for a detailed discussion of the use of α-points for machinescheduling problems.

Significant progress has recently been made in understanding the approximabilityof scheduling problems with the average weighted completion time objective. Skutellaand Woeginger [34] developed a polynomial-time approximation scheme for schedulingidentical parallel machines in the absence of release dates, P | |∑wjCj . Subsequently,several research groups have found polynomial-time approximation schemes for prob-lems with release dates such as P | rj |

∑wjCj and Rm | rj |

∑wjCj ; see the resulting

joint conference proceedings publication [1] for details.

We now briefly discuss some practical consequences of our work. Savelsbergh,Uma, and Wein [25] and Uma and Wein [38] performed experimental studies toevaluate, in part, the quality of the LP relaxation and approximation algorithmsstudied herein for 1| rj |

∑wjCj and related scheduling problems. The first authors

report that, except for instances that were deliberately constructed to be hard forthis approach, the present formulation and algorithms “deliver surprisingly strongexperimental performance.” They also note that “the ideas that led to improvedapproximation algorithms also lead to heuristics that are quite effective in empiricalexperiments; furthermore [. . . ] they can be extended to give improved heuristics formore complex problems that arise in practice.” While the authors of the follow-upstudy [38] report that when coupled with local improvement the LP-based heuristicsgenerally produce the best solutions, they also find that a simple heuristic often out-performs the LP-based heuristics. Whenever the machine becomes idle, this heuristicstarts nonpreemptively processing an available job of largest wj/pj ratio. By an-alyzing the differences between the LP schedule and this heuristic schedule, Chou,Queyranne, and Simchi-Levi [7] have subsequently shown the asymptotic optimalityof this on-line heuristic for classes of instances with bounded job weights and boundedprocessing times.

The contents of this paper are as follows. Section 2 is concerned with the LPrelaxations and their relationship. We begin with a presentation and discussion of theLP schedule. In section 2.1 we then review a time-indexed formulation introduced byDyer and Wolsey [9] and show that it is solved to optimality by the LP schedule. Insection 2.2 we present the mean busy time relaxation (or completion time relaxation)and prove, among other properties, its equivalence to the time-indexed formulation.Section 2.3 explores some polyhedral consequences, in particular the fact that themean busy time relaxation is (up to scaling by the job processing times) a super-modular linear program and that the “job-based” method for constructing the LP


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4 3 42

12

4

r4 r3 r2 r1

Fig. 1. An LP schedule for a 4-job instance given by r1 = 11, p1 = 1, r2 = 7, p2 = 5,r3 = 2, p3 = 3, r4 = 0, p4 = 5. Higher rectangles represent jobs with larger weight toprocessing time ratio. Time is shown on the horizontal axis.

schedule is equivalent to the corresponding greedy algorithm. Section 3 then dealswith approximation algorithms derived from these LP relaxations. In section 3.1 wepresent a method for constructing (αj)-schedules, which allows us to analyze andbound the job completion times in the resulting schedules. In section 3.2 we derivesimple bounds for α-schedules and (αj)-schedules, using a deterministic common α oruniformly distributed random αj ’s. Using appropriate probability distributions, weimprove the approximation bound to the value of 1.7451 for α-schedules in section 3.3and to the value of 1.6853 for (αj)-schedules in section 3.4. We also indicate howthese algorithms can be derandomized in O(n2) time for constructing deterministicschedules with these performance guarantees. In section 3.5 we show that our ran-domized approximations also apply in an on-line setting, and in section 3.6 we presenta class of “bad” instances for which the ratio of the optimal objective function valueand our LP bound is arbitrarily close to e

e−1 ≈ 1.5819. This constant defines a lowerbound on the approximation results that can be obtained by the present approach.We conclude in section 4 by discussing some related problems and open questions.

2. Relaxations. In this section, we present two linear programming relaxationsfor 1| rj |

∑wjCj . We show their equivalence and discuss some polyhedral conse-

quences.For both relaxations, the following preemptive schedule plays a crucial role: at any

point in time, schedule (preemptively) the available job with the highest wj/pj ratio.We assume (throughout the paper) that the jobs are indexed in order of nonincreasingratios w1

p1≥ w2

p2≥ · · · ≥ wn

pn, and ties are broken according to this order. Therefore,

whenever a job is released, the job being processed (if any) is preempted if the releasedjob has a smaller index. We refer to this preemptive schedule as the LP schedule. SeeFigure 1 for an example of an LP schedule.

Notice that this LP schedule does not in general minimize∑

j wjCj over allpreemptive schedules. This should not be surprising since the preemptive problem1| rj , pmtn |∑wjCj is (strongly) NP-hard [16]. It can be shown, however, that thetotal weighted completion time of the LP schedule is always within a factor of 2 ofthe optimal value for 1| rj , pmtn |∑wjCj , and this bound is tight; see [29].

The LP schedule can be constructed in O(n log n) time. To see this, we nowdescribe an implementation, which may be seen as “dynamic” (event-oriented) or,using the terminology of [19], “machine-based,” and can even be executed on-linewhile the jobs dynamically arrive over time. The algorithm keeps a priority queue [8]of the currently available jobs that have not yet been completely processed, with theratio wj/pj as the key and with another field indicating the remaining processing time.A scheduling decision is made at only two types of events: when a job is released and


when a job completes its processing. In the former case, the released job is added tothe priority queue. In the latter case, the completed job is removed from the priorityqueue. Then, in either case, the top element of the priority queue (the one with thehighest wj/pj ratio) is processed; if the queue is empty, then move on to the nextjob release; if there is none, then all jobs have been processed and the LP schedule iscomplete. This implementation results in a total of O(n) priority queue operations.Since each such operation can be implemented in O(logn) time [8], the algorithm runsin O(n log n) time.

The LP schedule can also be defined in a somewhat different manner, which maybe seen as “static” or “job-based” [19]. Consider the jobs one at a time in orderof nonincreasing wj/pj . Schedule each job j as early as possible starting at rj andpreempting it whenever the machine is busy processing another job (that thus cameearlier in the wj/pj ordering). This point of view leads to an alternate O(n log n)construction of the LP schedule; see [10].

2.1. Time-indexed relaxation. Dyer and Wolsey [9] investigate several typesof relaxations of 1| rj |

∑wjCj , the strongest ones being time-indexed. We consider

the weaker of their two time-indexed formulations, which they call formulation (D).It uses two types of variables: yjτ = 1 if job j is being processed during time in-terval [τ, τ + 1), and zero otherwise; and tj represents the start time of job j. Forsimplicity, we add pj to tj and replace the resulting expression by Cj ; this gives anequivalent relaxation.

ZD = min∑j∈N

wjCj

subject to

(D)∑

j:rj≤τ

yjτ ≤ 1, τ = 0, 1, . . . , T − 1,

T−1∑τ=rj

yjτ = pj , j ∈ N,

Cj =1

2pj +

1

pj

T−1∑τ=rj

(τ +

1

2

)yjτ , j ∈ N,(2.1)

0 ≤ yjτ , j ∈ N, τ = rj , . . . , T − 1,

where T is an upper bound on the makespan of an optimal schedule. (For example,T = maxj∈N rj+

∑j∈N pj .) We refer to this relaxation as the preemptive time-indexed

relaxation. The expression for Cj given in (2.1) corresponds to the correct value ofthe completion time if job j is not preempted; an interpretation in terms of meanbusy times is given in the next section for the case of preemptions. Observe thatthe number of variables of this formulation is pseudopolynomial. If we eliminate Cj

from the relaxation by using (2.1), we obtain a transportation problem [9] and, as aresult, yjτ can be assumed to be integral.

Lemma 2.1. There exists an optimal solution to (D) for which yjτ ∈ {0, 1} forall j and τ .

As indicated in [9], (D) can be solved in O(n log n) time. Actually, one can derivea feasible solution to (D) from the LP schedule by letting yLPjτ be equal to 1 if job jis being processed in [τ, τ + 1), and 0 otherwise.


Theorem 2.2. The solution yLP derived from the LP schedule is an optimalsolution to (D).

Proof. The proof is based on an interchange argument. Consider any optimal 0/1-solution y∗ to (D). If there exist j < k and σ > τ ≥ rj such that y∗jσ = y∗kτ = 1,then by replacing y∗jσ and y∗kτ by 0, and y∗jτ and y∗kσ by 1, we obtain another feasible

solution with an increase in the objective function value of (σ − τ)(wk

pk− wj

pj) ≤ 0.

The resulting solution must therefore also be optimal. By repeating this interchangeargument, we derive that there exists an optimal solution y∗ such that there do notexist j < k and σ > τ ≥ rj such that y∗jσ = y∗kτ = 1. This implies that the solution y∗

must correspond to the LP schedule.In particular, despite the pseudopolynomial number of variables in the LP Re-

laxation (D) an optimal solution can be obtained efficiently. We will make use ofthis fact as well as of the special structure of the LP schedule in the design andanalysis of the approximation algorithms; see section 3. We note again that in spiteof its nice properties the preemptive time-indexed LP Relaxation (D) solves neither1| rj |

∑wjCj nor 1| rj , pmtn |∑wjCj . In the former case, the processing of a job in

the LP solution may fail to be consecutive; in the latter case (2.1) does not necessarilydefine the completion time of a job in the preemptive LP schedule, as will be shownin the next lemma.

2.2. Mean busy time relaxation. Given any preemptive schedule, let Ij bethe indicator function of the processing of job j at time t; i.e., Ij(t) is 1 if the machineis processing j at time t, and 0 otherwise. To avoid pathological situations, we requirethat, in any preemptive schedule, when the machine starts processing a job, it doesso for a positive amount of time. Given any preemptive schedule, we define the meanbusy time Mj of job j to be the average time at which the machine is processing j,that is,

Mj :=1

pj

∫ T

rj

Ij(t) t dt.

For instance, in the example given in Figure 1, which will be used throughout thispaper, the mean busy time of job 4 is 5.5.

We first establish some important properties of Mj in the next two lemmas.Lemma 2.3. For any preemptive schedule, let Cj and Mj denote the completion

and mean busy time, respectively, of job j. Then for any job j, we haveMj+12pj ≤ Cj,

with equality if and only if job j is not preempted.Proof. If job j is processed without preemption, then Ij(t) = 1 if Cj − pj ≤ t <

Cj , and 0 otherwise; therefore, Mj +12pj = Cj . Otherwise, job j is not processed

during some interval(s) of total length L > 0 between times Cj − pj and Cj . Since∫ T

rjIj(t) dt = pj , job j must be processed during some time interval(s) of the same

total length L before Cj − pj . Therefore,

Mj =1

pj

∫ Cj

rj

Ij(t) t dt <1

pj

∫ Cj

Cj−pj

t dt = Cj − 1

2pj

and the proof is complete.Let S ⊆ N denote a set of jobs and define

p(S) :=∑j∈S

pj and rmin(S) := minj∈S

rj .


Let IS(t) :=∑

j∈S Ij(t). Since, by the machine capacity constraint, IS(t) ∈ {0, 1} forall t, we may view IS as the indicator function for job set S. We can thus define the

mean busy time of set S as MS := 1p(S)

∫ T

0IS(t) t dt. Note that we have

p(S)MS =

∫ T

0

∑j∈S

Ij(t)

t dt =∑j∈S

∫ T

0

Ij(t) t dt =∑j∈S

pjMj .(2.2)

So, unlike its start and completion time, the mean busy time of a job set is a simpleweighted average of the mean busy times of its elements. One consequence of thisobservation is the validity of the shifted parallel inequalities (2.3) (see, e.g., [10, 23, 24])below.

Lemma 2.4. For any set S of jobs and any preemptive schedule with mean busytime vector M , we have∑

j∈SpjMj ≥ p(S)

(rmin(S) +

1

2p(S)

),(2.3)

and equality holds if and only if all jobs in S are scheduled without interruptionfrom rmin(S) to rmin(S) + p(S).

Proof. Note that∑

j∈S pjMj = p(S)MS =∫ T

rmin(S)IS(t) t dt, that IS(t) = 0

for t < rmin(S) and IS(t) ≤ 1 for t ≥ rmin(S), and that∫ T

rmin(S)IS(t)dt = p(S). Under

these constraints, MS is minimized when IS(t) = 1 for rmin(S) ≤ t < rmin(S) + p(S),and 0 otherwise. Therefore, MS is uniquely minimized among all feasible preemptiveschedules when all jobs in S are continuously processed from rmin(S) to rmin(S)+p(S).This minimal value is p(S)(rmin(S) +

12p(S)) and is a lower bound for

∑j∈S pjMj in

any feasible preemptive schedule.As a result of Lemma 2.4, the following linear program provides a lower bound

on the optimal value of 1| rj , pmtn |∑wjCj , and hence on that of 1| rj |∑

wjCj .

ZR = min∑j∈N

wj

(Mj +

1

2pj

)(R) subject to ∑

j∈SpjMj ≥ p(S)

(rmin(S) +

1

2p(S)

), S ⊆ N.

The proof of the following theorem and later developments use the notion ofcanonical decompositions [10]. For a set S of jobs, consider the schedule which pro-cesses jobs in S as early as possible, say, in order of their release dates. This scheduleinduces a partition of S into {S1, . . . , Sk} such that the machine is busy processingjobs in S exactly in the disjoint intervals [rmin(S�), rmin(S�) + p(S�)] for = 1, . . . , k.We refer to this partition as the canonical decomposition of S. A set is canonical if itis identical to its canonical decomposition, i.e., if its canonical decomposition is {S}.Thus a set S is canonical if and only if it is feasible to schedule all its jobs in the timeinterval [rmin(S), rmin(S)+ p(S)). Note that the set N = {1, 2, 3, 4} in our example iscanonical, whereas the subset {1, 2, 3} is not; it has the decomposition {{3}, {1, 2}}.Let

h(S) :=

k∑�=1

p(S�)

(rmin(S�) +

1

2p(S�)

),(2.4)


where {S1, . . . , Sk} is the canonical decomposition of S ⊆ N . Then Lemma 2.4

implies that∑

j∈S pjMj =∑k

�=1

∑j∈S�

pjMj ≥ h(S) is a valid inequality for themean busy time vector of any preemptive schedule. In other words, Relaxation (R)may be written as

min

∑j∈N

wj

(Mj +

1

2pj

):∑j∈S

pjMj ≥ h(S) for all S ⊆ N

.

Theorem 2.5. Let MLPj be the mean busy time of job j in the LP schedule.

Then MLP is an optimal solution to (R).Proof. By Lemma 2.4, MLP is a feasible solution for (R).To prove optimality of MLP , we construct a lower bound on the optimal value

of (R) and show that it is equal to the objective function value of MLP . Recall thatthe jobs are indexed in nonincreasing order of the wj/pj ratios; let [i] := {1, 2, . . . , i},and let Si

1, . . . , Sik(i) denote the canonical decomposition of [i]. Observe that for any

vector M = (Mj)j∈N we have

∑j∈N

wjMj =

n∑i=1

(wi

pi− wi+1

pi+1

)∑j∈[i]

pjMj =

n∑i=1

(wi

pi− wi+1

pi+1

) k(i)∑�=1

∑j∈Si

�

pjMj ,(2.5)

where we let wn+1/pn+1 := 0. We have therefore expressed∑

j∈N wjMj as a non-negative combination of expressions

∑j∈Si

�pjMj over canonical sets. By construction

of the LP schedule, the jobs in any of these canonical sets Si� are continuously pro-

cessed from rmin(Si�) to rmin(S

i�) + p(Si

�) in the LP schedule. Thus, for any feasiblesolution M to (R) and any such canonical set Si

� we have∑j∈Si

�

pjMj ≥ h(Si�) = p(Si

�)

(rmin(S

i�) +

1

2p(Si

�)

)=∑j∈Si

�

pjMLPj ,

where the last equation follows from Lemma 2.4. Combining this with (2.5), we derivea lower bound on

∑j wjMj for any feasible solution M to (R), and this lower bound

is attained by the LP schedule.From Theorems 2.2 and 2.5, we derive that the values of the two Relaxations (D)

and (R) are equal.Corollary 2.6. The LP Relaxations (D) and (R) of 1 | rj |

∑wjCj yield the

same optimal objective function value, i.e., ZD = ZR, for any weights w ≥ 0. Thisvalue can be computed in O(n log n) time.

Proof. For the equivalence of the two lower bounds, note that the mean busytime MLP

j of any job j in the LP schedule can be expressed as

MLPj =

1

pj

T−1∑τ=rj

yLPjτ

(τ +

1

2

),(2.6)

where yLP is the solution to (D) derived from the LP schedule. The result then followsdirectly from Theorems 2.2 and 2.5. We have shown earlier that the LP schedule canbe constructed in O(n log n) time.

Although the LP schedule does not necessarily minimize the objective func-tion

∑j wjCj over the preemptive schedules, Theorem 2.5 implies that it minimizes

174 GOEMANS, QUEYRANNE, SCHULZ, SKUTELLA, AND WANG∑j wjMj over the preemptive schedules. In addition, by Lemma 2.3, the LP schedule

is also optimal for both preemptive and nonpreemptive problems 1| rj , pmtn |∑wjCj ,and 1| rj |

∑wjCj whenever it does not preempt any job. For example, this is the case

if all processing times are equal to 1 or if all jobs are released at the same date. Thus,the LP schedule provides an optimal solution to problems 1| rj , pj = 1 |∑wjCj andto 1| |∑wjCj . This was already known. In the latter case it coincides with Smith’sratio rule [35]; see Queyranne and Schulz [24] for the former case.

2.3. Polyhedral consequences. We now consider some polyhedral consequen-ces of the preceding results. Let P∞

D be the feasible region defined by the constraintsof Relaxation (D) when T = ∞, i.e.,

P∞D :=

y ≥ 0 :∑

j:rj≤τ

yjτ ≤ 1 for τ ∈ N;∑τ≥rj

yjτ = pj for all j ∈ N

.

In addition, we denote by PR := {M ∈ RN :

∑j∈S pjMj ≥ h(S) for all S ⊆ N} the

polyhedron defined by the constraints of Relaxation (R).Theorem 2.7.(i) Polyhedron PR is the convex hull of the mean busy time vectors M of all

preemptive schedules. Moreover, every vertex of PR is the mean busy timevector of an LP schedule.

(ii) Polyhedron PR is also the image of P∞D in the space of the M -variables under

the linear mapping M : y → M(y) ∈ RN defined by

M(y)j =1

pj

∑τ≥rj

yjτ

(τ +

1

2

)for all j ∈ N.

Proof. (i) Lemma 2.4 implies that the convex hull of the mean busy time vectorsMof all feasible preemptive schedules is contained in PR. To show the reverse inclusion,it suffices to show that (a) every extreme point of PR corresponds to a preemptiveschedule; and (b) every extreme ray of PR is a direction of recession for the convexhull of mean busy time vectors. Property (a) and the second part of statement (i)follow from Theorem 2.5 and the fact that every extreme point of PR is the uniqueminimizer of

∑j∈N wjMj for some w ≥ 0. For (b), note that the extreme rays of PR

are the n unit vectors of RN . An immediate extension to preemptive schedules and

mean busy times of results in Balas [3] implies that these unit vectors of RN are

directions of recession for the convex hull of mean busy time vectors. This completesthe proof of (i).

(ii) We first show that the image M(P∞D ) of P∞

D is contained in PR. For this,let y be a vector in P∞

D and S ⊆ N with canonical decomposition {S1, . . . , Sk}. Bydefinition of M(y)j , we have∑

j∈SpjM(y)j =

∑j∈S

∑τ≥rj

yjτ

(τ +

1

2

)

≥k∑

�=1

rmin(S�)+p(S�)−1∑τ=rmin(S�)

(τ +

1

2

)

=k∑

�=1

p(S�)

(rmin(S�) +

1

2p(S�)

)= h(S).


The inequality follows from the constraints defining P∞D and the interchange argument

which we already used in the proof of Theorem 2.2. This shows M(y) ∈ PR and thusM(P∞

D ) ⊆ PR.To show the reverse inclusion, we use the observation from the proof of part (i)

that PR can be represented as the sum of the convex hull of the mean busy timevectors of all LP schedules and the nonnegative orthant. Since, by (2.6), the meanbusy time vector MLP of any LP schedule is the projection of the corresponding 0/1-vector yLP , it remains to show that every unit vector ej is a direction of recessionfor M(P∞

D ). For this, fix an LP schedule and let yLP and MLP = M(yLP ) denote theassociated 0/1 y-vector and mean busy time vector, respectively. For any job j ∈ Nand any real λ > 0, we need to show that MLP + λ ej ∈ M(P∞

D ).Let τmax := argmax{yLPkτ = 1 : k ∈ N}. Choose θ such that yLPjθ = 1, choose an

integer µ > max{λ pj , τmax − θ}, and define y′ by y′jθ = 0, y′j,θ+µ = 1, and y′kτ = yLPkτotherwise. In the associated preemptive schedule, the processing of job j that was donein interval [θ, θ+1) is now postponed, by µ time units, until interval [θ+µ, θ+µ+1).Therefore, its mean busy time vector M ′ = M(y′) satisfies M ′

j = MLPj + µ/pj and

M ′k = MLP

k for all k �= j. Let λ′ := µ/pj ≥ λ, so M ′ = MLP + λ′ej . Then the vectorMLP + λ ej is a convex combination of MLP = M(yLP ) and M ′ = M(y′). Let ybe the corresponding convex combination of yLP and y′. Since P∞

D is convex, theny ∈ P∞

D and, since the mapping M is linear, MLP + λ ej = M(y) ∈ M(P∞D ).

In view of earlier results for single machine scheduling with identical releasedates [22], as well as for parallel machine scheduling with unit processing times andinteger release dates [24], it is interesting to note that the feasible set PR of the meanbusy time relaxation is, up to scaling by the job processing times, a supermodularpolyhedron.

Proposition 2.8. The set function h defined in (2.4) is supermodular.Proof. Consider any two elements j, k ∈ N and any subset S ⊆ N \ {j, k}. We

may construct an LP schedule minimizing∑

i∈S∪{k} piMi using the job-based method

by considering first the jobs in S and then job k. (Note that considering the jobs in anysequence leads to a schedule minimizing

∑i piMi because jobs are weighted by their

processing times in this objective function.) By Definition (2.4) the resulting meanbusy times MLP satisfy

∑i∈S piM

LPi = h(S) and

∑i∈S∪{k} piM

LPi = h(S ∪ {k}).

Note that job k is scheduled, no earlier than its release date, in the first pk units ofidle time left after the insertion of all jobs in S. Thus MLP

k is the mean of all these pktime units. Similarly, we may construct an LP schedule, whose mean busy time vector

will be denoted by MLP , minimizing∑

i∈S∪{j, k} piMi by considering first the jobs in

S, so MLPi = MLP

i for all i ∈ S; then job j, so∑

i∈S∪{j} piMLPi = h(S ∪ {j}); and

then job k, so∑

i∈S∪{j, k} piMLPi = h(S ∪{j, k}). Since job j has been inserted after

subset S was scheduled, job k cannot use any idle time interval that is earlier thanthose it used in the former LP schedule MLP—and some of the previously availableidle time may now be occupied by job j, causing a delay in the mean busy time ofjob k; thus we have MLP

k ≥ MLPk and therefore

h(S ∪ {j, k})− h(S ∪ {j}) = pkMLPk ≥ pkM

LPk = h(S ∪ {k})− h(S).

This suffices to establish that h is supermodular.An alternate proof of the supermodularity of h can be derived, as in [10], from the

fact, observed by Dyer and Wolsey and already mentioned above, that Relaxation (D)becomes a transportation problem after elimination of the Cj ’s. Indeed, from an


interpretation of Nemhauser, Wolsey, and Fisher [20] of a result of Shapley [31],it then follows that the value of this transportation problem as a function of S issupermodular. One of the consequences of Proposition 2.8 is that the job-basedmethod to construct an LP schedule is just a manifestation of the greedy algorithmfor minimizing

∑j∈N wjMj over the supermodular polyhedron PR.

We finally note that the separation problem for the polyhedron PR can be solvedcombinatorially. One can separate over the family of inequalities

∑j∈S pjMj ≥

p(S)(rmin(S) + p(S)) by trying all possible values for rmin(S) (of which there are atmost n) and then applying a O(n log n) separation routine of Queyranne [22] for theproblem without release dates. The overall separation routine can be implemented inO(n2) time by observing that the bottleneck step in Queyranne’s algorithm—sortingthe mean busy times of the jobs—needs to be done only once for the whole job set.

3. Provably good schedules and LP relaxations. In this section, we deriveapproximation algorithms for 1| rj |

∑wjCj that are based on converting the preemp-

tive LP schedule into a feasible nonpreemptive schedule whose value can be boundedin terms of the optimal LP value ZD = ZR. This yields results on the quality of boththe computed schedule and the LP relaxations under consideration since the value ofthe computed schedule is an upper bound and the optimal LP value is a lower boundon the value of an optimal schedule.

In section 3.6, we describe a family of instances for which the ratio between theoptimal value of the 1| rj |

∑wjCj problem and the lower bounds ZR and ZD is

arbitrarily close to ee−1 > 1.5819. This lower bound of e

e−1 sets a target for the designof approximation algorithms based on these LP relaxations.

In order to convert the preemptive LP schedule into a nonpreemptive schedule wemake use of so-called α-points of jobs. For 0 < α ≤ 1 the α-point tj(α) of job j is thefirst point in time when an α-fraction of job j has been completed in the LP schedule,i.e., when j has been processed for αpj time units. In particular, tj(1) is equal to thecompletion time and we define tj(0

+) to be the start time of job j. Notice that, bydefinition, the mean busy time MLP

j of job j in the LP schedule is the average overall its α-points

MLPj =

∫ 1

0

tj(α) dα.(3.1)

We will also use the following notation: For a fixed job j and 0 < α ≤ 1 we denotethe fraction of job k that is completed in the LP schedule by time tj(α) by ηk(α); inparticular, ηj(α) = α. The amount of idle time that occurs between time 0 and thestart of job j in the LP schedule is denoted by τidle. Note that ηk and τidle implicitlydepend on the fixed job j. By construction, there is no idle time between the startand completion of job j in the LP schedule; therefore we can express j’s α-point as

tj(α) = τidle +∑k∈N

ηk(α)pk.(3.2)

For a given 0 < α ≤ 1, we define the α-schedule as the schedule in which jobsare processed nonpreemptively as early as possible and in the order of nondecreasingα-points. We denote the completion time of job j in this schedule by Cα

j . The ideaof scheduling nonpreemptively in the order of α-points in a preemptive schedule wasintroduced by Phillips, Stein, and Wein [21] and used in many of the subsequentresults in the area.


t2(9/10)

2

42

1

4

3 42

1

3

t4(1/5)

4

t3(1/2)

t4(1/2)

t2(1/2)

t1(1/2)

t1(1/2)

t3(2/3)

21

4 3

Fig. 2. A nonpreemptive α-schedule (for α = 1/2) and an (αj)-schedule shown aboveand below the LP schedule, respectively. Notice that there is no common α value that wouldlead to the latter schedule.

This idea can be further extended to individual, i.e., job-dependent αj-pointstj(αj), for j ∈ N and 0 < αj ≤ 1. We denote the vector consisting of all αj ’s byααα := (αj) := (α1, . . . , αn). Then, the (αj)-schedule is constructed by processing thejobs as early as possible and in nondecreasing order of their αj-points; the completiontime of job j in the (αj)-schedule is denoted by C ααα

j . Figure 2 compares an α-scheduleto an (αj)-schedule both derived from the LP schedule in Figure 1.

In what follows we present several results on the quality of α-schedules and (αj)-schedules. These results also imply bounds on the quality of the LP relaxationsdiscussed in the previous section. The main result is the construction of a random(αj)-schedule whose expected value is at most a factor 1.6853 of the optimal LP valueZD = ZR. Therefore, the LP Relaxations (D) and (R) deliver a lower bound which isat least 0.5933 (≈ 1.6853−1) times the optimal value. The corresponding randomizedalgorithm can be implemented on-line; it has competitive ratio 1.6853 and runningtime O(n log n); it can also be derandomized to run off-line in O(n2) time. We alsoinvestigate the case of a single common α and show that the best α-schedule is alwayswithin a factor of 1.7451 of the optimum.

3.1. Bounding the completion times in (αj)-schedules. To analyze thecompletion times of jobs in (αj)-schedules, we consider nonpreemptive schedules ofsimilar structure that are, however, constructed by a slightly different conversionroutine which we call (αj)-Conversion.

Consider the jobs j ∈ N in order of nonincreasing αj-points tj(αj)


3

42

t2(9/10)

21

t4(1/5)

34

2

4

1

4

4

t1(1/2)

t3(2/3)

2

4

3 4

1

24

4

3

4

4

1

4

23

1

4

Fig. 3. Illustration of the individual iterations of (αj)-Conversion.

and iteratively change the preemptive LP schedule to a nonpreemp-tive schedule by applying the following steps:(i) remove the αj pj units of job j that are processed before tj(αj)

and leave the machine idle during the corresponding time inter-vals; we say that this idle time is caused by job j;

(ii) delay the whole processing that is done later than tj(αj) by pj ;(iii) remove the remaining (1−αj)-fraction of job j from the machine

and shrink the corresponding time intervals; shrinking a timeinterval means to discard the interval and move earlier, by thecorresponding amount, any processing that occurs later;

(iv) process job j in the released time interval [tj(αj), tj(αj) + pj).

Figure 3 contains an example illustrating the action of (αj)-Conversion startingfrom the LP schedule of Figure 1. Observe that in the resulting schedule jobs areprocessed in nondecreasing order of αj-points, and no job j is started before timetj(αj) ≥ rj . The latter property will be useful in the analysis of on-line (αj)-schedules.


Lemma 3.1. The completion time of job j in the schedule constructed by (αj)-Conversion is equal to

tj(αj) +∑

kαk≤ηk(αj)

(1 + αk − ηk(αj)

)pk.

Proof. Consider the schedule constructed by (αj)-Conversion. The completiontime of job j is equal to the idle time before its start plus the sum of processing timesof jobs that start no later than j. Since the jobs are processed in nondecreasing orderof their αj-points, the amount of processing before the completion of job j is∑

kαk≤ηk(αj)

pk.(3.3)

The idle time before the start of job j can be written as the sum of the idle time τidlethat already existed in the LP schedule before j’s start plus the idle time before thestart of job j that is caused in step (i) of (αj)-Conversion; notice that step (iii) doesnot create any additional idle time since we shrink the affected time intervals. Eachjob k that is started no later than j, i.e., such that ηk(αj) ≥ αk, contributes αk pkunits of idle time; all other jobs k only contribute ηk(αj) pk units of idle time. As aresult, the total idle time before the start of job j can be written as

τidle +∑

kαk≤ηk(αj)

αk pk +∑

kαk>ηk(αj)

ηk(αj) pk.(3.4)

The completion time of job j in the schedule constructed by (αj)-Conversion isequal to the sum of the expressions in (3.3) and (3.4); the result then follows from(3.2).

It follows from Lemma 3.1 that the completion time Cj of each job j in thenonpreemptive schedule constructed by (αj)-Conversion satisfies Cj ≥ tj(αj)+pj ≥rj+pj ; hence is a feasible schedule. Since the (αj)-schedule processes the jobs as earlyas possible and in the same order as the (αj)-Conversion schedule, we obtain thefollowing corollary.

Corollary 3.2. The completion time of job j in an (αj)-schedule can be boundedby

C αααj ≤ tj(αj) +

∑k

αk≤ηk(αj)

(1 + αk − ηk(αj)

)pk.

3.2. Bounds for α-schedules and (αj)-schedules. We start with a result onthe quality of the α-schedule for a fixed common value of α.

Theorem 3.3. For fixed α, (i) the value of the α-schedule is within a factormax

{1 + 1

α , 1 + 2α}of the optimal LP value; in particular, for α = 1/

√2 the bound

is 1+√2. Simultaneously, (ii) the length of the α-schedule is within a factor of 1+α

of the optimal makespan.Proof. While the proof of (ii) is an immediate consequence of (3.2) and Corol-

lary 3.2, it follows from the proof of Theorem 2.5 that for (i) it is sufficient to provethat, for any canonical set S, we have

∑j∈S

pjCαj ≤ max

(1 +

1

α, 1 + 2α

)p(S)

(rmin(S) +

1

2p(S)

)+

1

2

∑j∈S

p2j

.(3.5)


Indeed, using (2.5) and Lemma 2.4 it would then follow that

∑j∈N

wjCαj =

n∑i=1

(wi

pi− wi+1

pi+1

) k(i)∑�=1

∑j∈Si

�

pjCαj

≤ max

(1 +

1

α, 1 + 2α

) n∑i=1

(wi

pi− wi+1

pi+1

) k(i)∑�=1

∑j∈Si

�

pj

(MLP

j +pj2

)= max

(1 +

1

α, 1 + 2α

)∑j∈N

wj

(MLP

j +pj2

)= max

(1 +

1

α, 1 + 2α

)ZR,

thus proving the result.

Now consider any canonical set S and let us assume that, after renumberingthe jobs, S = {1, 2, . . . , } and t1(α) < t2(α) < · · · < t�(α) (so the ordering is notnecessarily anymore in nonincreasing order of wj/pj). Now fix any job j ∈ S. FromCorollary 3.2, we derive that

Cαj ≤ tj(α) +

∑k:α≤ηk

(1 + α− ηk) pk,(3.6)

where ηk := ηk(α) represents the fraction of job k processed in the LP schedulebefore tj(α). Let R denote the set of jobs k such that tk(α) < rmin(S) (and thusα ≤ ηk). Since S is a canonical set, the jobs in S are processed continuously inthe LP schedule between rmin(S) and rmin(S) + p(S), and therefore every job k withα ≤ ηk is either in S or in R. Observe that

∑k∈R αpk ≤ rmin(S) implies that

p(R) ≤ 1α rmin(S). We can thus simplify (3.6) with

Cαj ≤ tj(α) +

1

αrmin(S) +

j∑k=1

(1 + α− ηk) pk.(3.7)

Since the jobs in S are scheduled with no gaps in [rmin(S), rmin(S) + p(S)], we havethat

tj(α) = rmin(S) +∑k∈S

ηkpk ≤ rmin(S) +

j∑k=1

ηkpk +

�∑k=j+1

αpk.(3.8)

Combining (3.7) and (3.8), we derive that

Cαj ≤

(1 +

1

α

)rmin(S) + αp(S) +

j∑k=1

pk.

Multiplying by pj and summing over S, we get


∑j∈S

pjCαj ≤

(1 +

1

α

)rmin(S)p(S) + αp(S)2 +

∑j∈S

j∑k=1

pjpk

=

(1 +

1

α

)rmin(S)p(S) +

(1

2+ α

)p(S)2 +

1

2

∑j∈S

p2j ,

which implies (3.5).

In what follows we will compare the completion time C αααj of every job j with its

“completion time” MLPj + 1

2pj in the LP schedule. However, for any fixed commonvalue of α, there exist instances which show that this type of job-by-job analysis cangive a bound no better than 1 +

√2 > 2.4142. One can also show that, for any

given value of α, there exist instances for which the objective function value of theα-schedule can be as bad as twice the LP lower bound.

In view of these results, it is advantageous to use several values of α, as it appearsthat no instance can be simultaneously bad for all choices of α. In fact, α-pointsdevelop their full power in combination with randomization, i.e., when a common α oreven job-dependent αj ’s are chosen randomly from (0, 1] according to an appropriatedensity function. This is also motivated by (3.1) which relates the expected α-pointof a job under a uniform distribution of α to the LP variable MLP

j . For randomvalues αj , we analyze the expected value of the resulting (αj)-schedule and compareit to the optimal LP value. Notice that a bound on the expected value proves theexistence of a vector (αj) such that the corresponding (αj)-schedule meets this bound.Moreover, for our results we can always compute such an (αj) in polynomial time byderandomizing our algorithms with standard methods; see Propositions 3.8 and 3.13.

Although the currently best known bounds can be achieved only for (αj)-scheduleswith job-dependent αj ’s, we investigate α-schedules with a single common α as well.On the one hand, this helps to better understand the potential advantages of (αj)-schedules; on the other hand, the randomized algorithm that relies on a single αadmits a natural derandomization. In fact, we can easily compute an α-schedule ofleast objective function value over all α between 0 and 1; we refer to this schedule asthe best-α-schedule. In Proposition 3.8 below, we will show that there are at mostn different α-schedules. The best-α-schedule can be constructed in O(n2) time byevaluating all these different schedules.

As a warm-up exercise for the kind of analysis we use, we start by proving abound of 2 on the expected worst-case performance ratio of uniformly generated (αj)-schedules in the following theorem. This result will then be improved by using moreintricate probability distributions and by taking advantage of additional insights intothe structure of the LP schedule.

Theorem 3.4. Let the random variables αj be pairwise independently and uni-formly drawn from (0, 1]. Then, the expected value of the resulting (αj)-schedule iswithin a factor 2 of the optimal LP value ZD = ZR.

Proof. Remember that the optimal LP value is given by∑

j wj(MLPj + 1

2pj). To

get the claimed result, we prove that EU [Cαααj ] ≤ 2(MLP

j + 12pj) for all jobs j ∈ N ,

where EU [F (ααα)] denotes the expectation of a function F of the random variable αααwhen the latter is uniformly distributed. The overall performance follows from thisjob-by-job bound by linearity of expectations.

Consider an arbitrary, but fixed job j ∈ N . To analyze the expected completiontime of j, we first keep αj fixed and consider the conditional expectation EU [C

αααj |αj ].

Since the random variables αj and αk are independent for each k �= j, Corollary 3.2


and (3.2) yield

EU [Cαααj |αj ] ≤ tj(αj) +

∑k �=j

pk

∫ ηk(αj)

0

(1 + αk − ηk(αj)

)dαk + pj

= tj(αj) +∑k �=j

(ηk(αj)− ηk(αj)

2

2

)pk + pj

≤ tj(αj) +∑k �=j

ηk(αj)pk + pj ≤ 2

(tj(αj) +

1

2pj

).

To obtain the unconditional expectation EU [Cαααj ] we integrate over all possible choices

of αj

EU [Cαααj ] =

∫ 1

0

EU [Cαααj |αj ] dαj ≤ 2

(∫ 1

0

tj(αj) dαj +1

2pj

)= 2

(MLP

j +1

2pj

);

the last equation follows from (3.1).

We turn now to deriving improved results. We start with an analysis of thestructure of the LP schedule. Consider any job j and assume that, in the LP schedule,j is preempted at time s and its processing resumes at time t > s. Then all jobs whichare processed between s and t have a smaller index; as a result, these jobs will becompletely processed between times s and t. Thus, in the LP schedule, between thestart time and the completion time of any job j, the machine is constantly busy,alternating between the processing of portions of j and the complete processing ofgroups of jobs with a smaller index. Conversely, any job preempted at the starttime tj(0

+) of job j will have to wait at least until job j is complete before itsprocessing can be resumed.

We capture this structure by partitioning, for a fixed job j, the set of jobs N \{j}into two subsets N1 and N2: Let N2 denote the set of all jobs that are processedbetween the start and completion of job j. All remaining jobs are put into subset N1.Notice that the function ηk is constant for jobs k ∈ N1; to simplify notation we writeηk := ηk(αj) for those jobs. For k ∈ N2, let 0 < µk < 1 denote the fraction of job jthat is processed before the start of job k; the function ηk is then given by

ηk(αj) =

{0 if αj ≤ µk,

1 if αj > µk

for k ∈ N2.

We can now rewrite (3.2) as

tj(αj) = τidle +∑k∈N1

ηkpk +∑k∈N2αj>µk

pk + αj pj = tj(0+) +

∑k∈N2αj>µk

pk + αj pj .

(3.9)

Plugging (3.9) into (3.1) yields

MLPj = tj(0

+) +∑k∈N2

(1− µk)pk +1

2pj ,(3.10)


and Corollary 3.2 can be rewritten as

C αααj ≤ tj(0

+) +∑k∈N1αk≤ηk

(1 + αk − ηk) pk +∑k∈N2αj>µk

(1 + αk) pk + (1 + αj) pj ,

(3.11)

where, for k ∈ N2, we have used the fact that αk ≤ ηk(αj) is equivalent to αj > µk.The expressions (3.9), (3.10), and (3.11) reflect the structural insights that we needfor proving stronger bounds for (αj)-schedules and α-schedules in what follows.

As mentioned above, the second ingredient for an improvement on the bound of 2is a more sophisticated probability distribution of the random variables αj . In viewof the bound on C ααα

j given in (3.11), we have to cope with two contrary phenomena:On the one hand, small values of αk keep the terms of the form (1 + αk − ηk) and(1 + αk) on the right-hand side of (3.11) small; on the other hand, choosing largervalues decreases the number of terms in the first sum on the right-hand side of (3.11).The balancing of these two effects contributes to reducing the bound on the expectedvalue of C ααα

j .

3.3. Improved bounds for α-schedules. In this subsection we prove the fol-lowing theorem.

Theorem 3.5. Let γ ≈ 0.4675 be the unique solution to the equation

1− γ2

1 + γ= γ + ln(1 + γ)

satisfying 0 < γ < 1. Define c := 1+γ1+γ−e−γ < 1.7451 and δ := 1− γ2

1+γ ≈ 0.8511. If αis chosen according to the density function

f(α) =

{(c− 1)eα if α ≤ δ,

0 otherwise,

then the expected value of the resulting random α-schedule is bounded by c times theoptimal LP value ZD = ZR.

Before we prove Theorem 3.5 we state two properties of the density function fthat are crucial for the analysis of the corresponding random α-schedule.

Lemma 3.6. The function f given in Theorem 3.5 is a density function with thefollowing properties:

(i)∫ η

0f(α)(1 + α− η) dα ≤ (c− 1) η for all η ∈ [0, 1],

(ii)∫ 1

µf(α)(1 + α) dα ≤ c (1− µ) for all µ ∈ [0, 1].

Property (i) is used to bound the delay of job j caused by jobs in N1, whichcorresponds to the first summation on the right-hand side of (3.11). The secondsummation reflects the delay of job j caused by jobs in N2 and will be bounded byproperty (ii).

Proof of Lemma 3.6. A short computation shows that δ = ln cc−1 . The function

f is a density function since∫ 1

0

f(α) dα = (c− 1)

∫ δ

0

eα dα = (c− 1)( c

c− 1− 1

)= 1.


In order to prove property (i), observe that for η ∈ [0, δ]∫ η

0

f(α)(1 + α− η) dα = (c− 1)

∫ η

0

eα(1 + α− η) dα = (c− 1)η.

For η ∈ (δ, 1] we therefore get∫ η

0

f(α)(1 + α− η) dα <

∫ δ

0

f(α)(1 + α− δ) dα = (c− 1)δ < (c− 1)η.

Property (ii) holds for µ ∈ (δ, 1] since the left-hand side is 0 in this case. For µ ∈ [0, δ]we have∫ 1

µ

f(α)(1 + α) dα = (c− 1)

∫ δ

µ

eα(1 + α) dα = (c− 1)(eδδ − eµµ)

= c

(1− γ2

1 + γ− eµ−γµ

1 + γ

)≤ c

(1− γ2 + (1 + µ− γ)µ

1 + γ

)= c

(1− (γ − µ)2 + (1 + γ)µ

1 + γ

)≤ c (1− µ).

This completes the proof of the lemma.Proof of Theorem 3.5. In Lemma 3.6, both (i) for η = 1 and (ii) for µ = 0 yield

Ef [α] ≤ c − 1, where Ef [α] denotes the expected value of a random variable α thatis distributed according to the density function f given in Theorem 3.5. Thus, usinginequality (3.11) and Lemma 3.6 we derive that

Ef

[Cαj

] ≤ tj(0+) + (c− 1)

∑k∈N1

ηkpk + c∑k∈N2

(1− µk)pk + c pj

≤ c tj(0+) + c

∑k∈N2

(1− µk)pk + c pj = c

(MLP

j +1

2pj

);

the last inequality follows from the definition of N1 and ηk, and the last equalityfollows from (3.10).

Notice that any density function satisfying properties (i) and (ii) of Lemma 3.6for some value c′ directly leads to the job-by-job bound Ef [C

αj ] ≤ c′

(MLP

j + 12pj

)for

the corresponding random α-schedule. It is easy to see that the unit function satisfiesLemma 3.6 with c′ = 2, which establishes the following variant of Theorem 3.4.

Corollary 3.7. Let the random variable α be uniformly drawn from (0, 1].Then, the expected value of the resulting α-schedule is within a factor 2 of the optimalLP value ZD = ZR.

The use of an exponential density function is motivated by the first propertyin Lemma 3.6; notice that the function α → (c − 1)eα satisfies it with equality.On the other hand, the exponential function is truncated in order to reduce the term∫ 1

µf(α)(1+α) dα in the second property. In fact, the truncated exponential function f

in Theorem 3.5 can be shown to minimize c′; it is therefore optimal for our analysis.In addition, there exists a class of instances for which the ratio of the expected cost ofan α-schedule, determined using this density function, to the cost of the optimal LPvalue is arbitrarily close to 1.745; this shows that the preceding analysis is essentiallytight in conjunction with truncated exponential functions.


Theorem 3.5 implies that the best-α-schedule has a value of at most 1.7451ZR.The following proposition shows that the randomized algorithm that yields the α-schedule can be easily derandomized because the sample space is small.

Proposition 3.8. There are at most n different α-schedules; they can be com-puted in O(n2) time.

Proof. As α goes from 0+ to 1, the α-schedule changes only whenever an α-point,say for job j, reaches a time at which job j is preempted. Thus, the total number ofchanges in the α-schedule is bounded from above by the total number of preemptions.Since a preemption can occur in the LP schedule only whenever a job is released, thetotal number of preemptions is at most n − 1, and the number of α-schedules is atmost n. Since each of these α-schedules can be computed in O(n) time, the result onthe running time follows.

3.4. Improved bounds for (αj)-schedules. In this subsection, we prove thefollowing theorem.

Theorem 3.9. Let γ ≈ 0.4835 be the unique solution to the equation

γ + ln(2− γ) = e−γ((2− γ)eγ − 1

)satisfying 0 < γ < 1. Define δ := γ+ln(2−γ) ≈ 0.8999 and c := 1+ e−γ/δ < 1.6853.Let the αj’s be chosen pairwise independently from a probability distribution over (0, 1]with the density function

g(α) =

{(c− 1)eα if α ≤ δ,

0 otherwise.

Then, the expected value of the resulting random (αj)-schedule is bounded by c timesthe optimal LP value ZD = ZR.

The bound in Theorem 3.9 also yields a bound on the quality of the LP relaxations.Corollary 3.10. The LP Relaxations (D) and (R) deliver in O(n log n) time a

lower bound which is at least 0.5933 (≈ 1.6853−1) times the objective function valueof an optimal schedule.

Following the lines of the last subsection, we state two properties of the densityfunction g that are crucial for the analysis of the corresponding random (αj)-schedule.

Lemma 3.11. The function g given in Theorem 3.9 is a density function with thefollowing properties:

(i)∫ η

0g(α)(1 + α− η) dα ≤ (c− 1) η for all η ∈ [0, 1],

(ii) (1 + Eg[α])∫ 1

µg(α) dα ≤ c (1− µ) for all µ ∈ [0, 1],

where Eg[α] denotes the expected value of a random variable α that is distributedaccording to g.

Notice the similarity of Lemma 3.11 and Lemma 3.6 of the last subsection. Again,properties (i) and (ii) are used to bound the delay of job j caused by jobs in N1 andN2, respectively, in the right-hand side of inequality (3.11). Property (i) for η = 1 orproperty (ii) for µ = 0 again yield Eg[α] ≤ c− 1.

Proof of Lemma 3.11. A short computation shows that δ = ln cc−1 . It thus follows

from the same arguments as in the proof of Lemma 3.6 that g is a density functionand that property (i) holds. In order to prove property (ii), we first compute

Eg[α] =

∫ 1

0

g(α)α dα = (c− 1)

∫ δ

0

eαα dα = c δ − 1.


Property (ii) certainly holds for µ ∈ (δ, 1]. For µ ∈ [0, δ] we get

(1 + Eg[α])

∫ 1

µ

g(α) dα = c δ (c− 1)

∫ δ

µ

eα dα

= c e−γ((2− γ)eγ − eµ

)= c (2− γ − eµ−γ)

≤ c(2− γ − (1 + µ− γ)

)= c (1− µ).

This completes the proof of the lemma.

Proof of Theorem 3.9. Our analysis of the expected completion time of job jin the random (αj)-schedule follows the line of argument developed in the proof ofTheorem 3.4. First we consider a fixed choice of αj and bound the correspondingconditional expectation Eg[C

αααj |αj ]. In a second step we bound the unconditional ex-

pectation Eg[Cαααj ] by integrating the product g(αj)Eg[C

αααj |αj ] over the interval (0, 1].

For a fixed job j and a fixed value αj , the bound in (3.11) and Lemma 3.11 (i)yield

Eg[Cαααj |αj ] ≤ tj(0

+) + (c− 1)∑k∈N1

ηk pk +∑k∈N2αj>µk

(1 + Eg[αk])pk + (1 + αj)pj

≤ c tj(0+) + (1 + Eg[α1])

∑k∈N2αj>µk

pk + (1 + αj)pj .

The last inequality follows from (3.9) and Eg[αk] = Eg[α1] for all k ∈ N . Usingproperty (ii) and (3.10) yields

Eg[Cαααj ] ≤ c tj(0

+) + (1 + Eg[α1])∑k∈N2

pk

∫ 1

µk

g(αj) dαj + (1 + Eg[αj ])pj

≤ c tj(0+) + c

∑k∈N2

(1− µk)pk + c pj = c

(MLP

j +1

2pj

).

The result follows from linearity of expectations.

While the total number of possible orderings of jobs is n! = 2O(n logn), we showin the following lemma that the maximal number of (αj)-schedules is at most 2n−1.We will use the following observation. Let qj denote the number of different pieces ofjob j in the LP schedule; thus qj represents the number of times job j is preemptedplus 1. Since there are at most n− 1 preemptions, we have that

∑nj=1 qj ≤ 2n− 1.

Lemma 3.12. The maximal number of (αj)-schedules is at most 2n−1, and this

bound can be attained.

Proof. The number of (αj)-schedules is given by s =∏n

j=1 qj . Notice that

q1 = 1 since this job is not preempted in the LP schedule. Thus, s =∏n

j=2 qj , while∑nj=2 qj ≤ 2(n− 1). By the arithmetic-geometric mean inequality, we have that

s =n∏

j=2

qj ≤(∑n

j=2 qj

n− 1

)n−1

≤ 2n−1.


Furthermore, this bound is attained if qj = 2 for j = 2, . . . , n, and this is achieved,for example, for the instance with pj = 2, wj = n − j + 1, and rj = n − j for allj.

Therefore, and in contrast to the case of random α-schedules, we cannot affordto derandomize the randomized 1.6853-approximation algorithm by enumerating all(αj)-schedules. We instead use the method of conditional probabilities [18].

From inequality (3.11) we obtain for every vector ααα = (αj) an upper bound onthe objective function value of the corresponding (αj)-schedule,

∑j wjC

αααj ≤ UB(ααα),

where UB(ααα) =∑

j wj RHSj(ααα) and RHSj(ααα) denotes the right-hand side of in-equality (3.11). Taking expectations and using Theorem 3.9, we have already shownthat

Eg

∑j∈N

wjCαααj

≤ Eg[UB(ααα)] ≤ cZD,

where c < 1.6853. For each job j ∈ N let Qj = {Qj1, . . . , Qjqj} denote the set ofintervals for αj corresponding to the qj pieces of job j in the LP schedule. We considerthe jobs one by one in arbitrary order, say, j = 1, . . . , n. Assume that, at step j ofthe derandomized algorithm, we have identified intervals Qd

1 ∈ Q1, . . . , Qdj−1 ∈ Qj−1

such that

Eg[UB(ααα) |αi ∈ Qdi for i = 1, . . . , j − 1] ≤ cZD .

Using conditional expectations, the left-hand side of this inequality is

Eg[UB(ααα) |αi ∈ Qdi for i = 1, . . . , j − 1]

=

qj∑�=1

Prob{αj ∈ Qj�}Eg

[UB(ααα) |αi ∈ Qd

i for i = 1, . . . , j − 1 and αj ∈ Qj�

].

Since∑qj

�=1 Prob{αj ∈ Qj�} = 1, there exists at least one interval Qj� ∈ Qj such that

Eg[UB(ααα) |αi ∈ Qdi for i = 1, . . . , j − 1 and αj ∈ Qj�]

≤ Eg[UB(ααα) |αi ∈ Qdi for i = 1, . . . , j − 1].

(3.12)

Therefore, it suffices to identify such an interval Qdj = Qj� satisfying (3.12), and we

may conclude that

Eg

[∑h∈N

whCαααh |αi ∈ Qd

i for i = 1, . . . , j

]≤ Eg[UB(ααα) |αi ∈ Qd

i for i = 1, . . . , j] ≤ cZD .

Having determined in this way an interval Qdj for every job j = 1, . . . , n, we then

note that the (αj)-schedule is the same for all ααα ∈ Qd1 × Qd

2 × · · · × Qdn. The (now

deterministic) objective function value of this (αj)-schedule is∑j∈N

wjCαααj ≤ Eg[UB(ααα) |αi ∈ Qd

i for i = 1, . . . , n]

≤ Eg[UB(ααα)] ≤ c ZD < 1.6853ZD ,


��

2

t1(1/2)

1

t4(1/5)

t3(2/3)

34

42

t2(9/10)

42

1

34

Fig. 4. The on-line schedule for the previously considered instance and αj-points. TheLP schedule is shown above for comparison.

as desired. For every j = 1, . . . , n, checking whether an interval Qdj satisfies in-

equality (3.12) amounts to evaluating O(n) terms, each of which may be computedin constant time. Since, as observed just before Lemma 3.12, we have a total of∑n

j=1 qj ≤ 2n − 1 candidate intervals, it follows that the derandomized algorithm

runs in O(n2) time.

Proposition 3.13. The randomized 1.6853-approximation algorithm can be de-randomized; the resulting deterministic algorithm runs in O(n2) time and has perfor-mance guarantee 1.6853 as well.

3.5. Constructing provably good schedules on-line. In this subsection weshow that our randomized approximation results also apply in an on-line setting.There are several different on-line paradigms that have been studied in the area ofscheduling; we refer to [30] for a survey. We consider the setting where jobs continuallyarrive over time, and, for each time t, we must construct the schedule until time twithout any knowledge of the jobs that will arrive afterwards. In particular, thecharacteristics of a job, i.e., its processing time and its weight, become known only atits release date.

It has already been shown in section 2 that the LP schedule can be constructed on-line. Unfortunately, for a given vector (αj), the corresponding (αj)-schedule cannotbe constructed on-line. We learn only about the position of a job k in the sequencedefined by nondecreasing αj-points at time tk(αk); therefore, we cannot start job kat an earlier point in time in the on-line setting. On the other hand, however, thestart time of k in the (αj)-schedule can be earlier than its αk-point tk(αk).

Although an (αj)-schedule cannot be constructed on-line, the above discussionreveals that the following variant, which we call the on-line-(αj)-schedule, can beconstructed on-line: For a given vector (αj), process the jobs as early as possible inthe order of their αj-points, with the additional constraint that no job k may startbefore time tk(αk). See Figure 4 for an example. We note that this idea of delayingthe start of jobs until sufficient information for a good decision is available was in thissetting introduced by Phillips, Stein, and Wein [21].


Notice that the nonpreemptive schedule constructed by (αj)-Conversion ful-fills these constraints; its value is therefore an upper bound on the value of the on-line-(αj)-schedule. Our analysis in the last subsections relies on the bound given inCorollary 3.2, which also holds for the schedule constructed by (αj)-Conversion byLemma 3.1. This yields the following results.

Theorem 3.14. For any instance of the scheduling problem 1| rj |∑

wjCj,

(a) choosing α = 1/√2 and constructing the on-line-α-schedule yields a determin-

istic on-line algorithm with competitive ratio 1 +√2 ≤ 2.4143 and running

time O(n log n);(b) choosing the αj’s randomly and pairwise independently from (0, 1] according

to the density function g of Theorem 3.9 and constructing the on-line-(αj)-schedule yields a randomized on-line algorithm with competitive ratio 1.6853and running time O(n log n).

The competitive ratio 1.6853 in Theorem 3.14 beats the deterministic on-line lowerbound 2 for the unit-weight problem 1| rj |

∑Cj [15, 36]. For the same problem,

Stougie and Vestjens [37] (see also [39]) proved the lower bound ee−1 > 1.5819 for

randomized on-line algorithms.

3.6. Bad instances for the LP relaxations. In this subsection, we describe afamily of instances for which the ratio between the optimal value of the 1| rj |

∑wjCj

problem and the lower bounds ZR and ZD is arbitrarily close to ee−1 > 1.5819.

These instances In have n ≥ 2 jobs as follows: one large job, denoted job n, andn− 1 small jobs, denoted j = 1, . . . , n− 1. The large job has processing time pn = n,weight wn = 1

n , and release date rn = 0. Each of the n − 1 small jobs j has zeroprocessing time, weight wj =

1n(n−1) (1 +

1n−1 )

n−j , and release date rj = j.

Throughout the paper, we have assumed that processing times are nonzero. Inorder to satisfy this assumption, we could impose a processing time of 1/k for all smalljobs, multiply all processing times and release dates by k to make the data integral,and then let k tend to infinity. For simplicity, however, we just let the processing timeof all small jobs be 0.

The LP solution has job n start at time 0, preempted by each of the small jobs;hence its mean busy times are MLP

j = rj for j = 1, . . . , n − 1 and MLPn = n

2 . Its

objective function value is ZR = (1 + 1n−1 )

n − (1 + 1n−1 ). Notice that the completion

time of each job j is in fact equal to MLPj + 1

2pj such that the actual value of thepreemptive schedule is equal to ZR.

Now consider an optimal nonpreemptive schedule C∗ and let k = �C∗n� − n ≥ 0,

so k is the number of small jobs that can be processed before job n. It is then optimalto process all these small jobs 1, . . . , k at their release dates and to start processingjob n at date rk = k just after job k. It is also optimal to process all remaining jobsk + 1, . . . , n− 1 at date k + n just after job n. Let Ck denote the resulting schedule;that is, Ck

j = j for all j ≤ k, and Ckj = k + n otherwise. Its objective function value

is (1+ 1n−1 )

n− 1n−1 − k

n(n−1) . Therefore, the optimal schedule is Cn−1 with objective

function value (1 + 1n−1 )

n − 1n−1 − 1

n . As n grows large, the LP objective functionvalue approaches e− 1 while the optimal nonpreemptive cost approaches e.

4. Conclusion. Even though polynomial-time approximation schemes have nowbeen discovered for the problem 1| rj |

∑wjCj [1], the algorithms we have developed,

or variants of them, are likely to be superior in practice. The experimental studiesof Savelsbergh, Uma, and Wein [25] and Uma and Wein [38] indicate that LP-based


relaxations and scheduling in order of αj-points are powerful tools for a variety ofscheduling problems.

Several intriguing questions remain open. Regarding the quality of linear pro-gramming relaxations, it would be interesting to close the gap between the upper(1.6853) and lower (1.5819) bound on the quality of the relaxations considered inthis paper. We should point out that the situation for the strongly NP-hard [16]problem 1| rj , pmtn |∑wjCj is similar. It is shown in [29] that the completion timerelaxation is in the worst case at least a factor of 8/7 and at most a factor of 4/3 offthe optimum; the latter bound is achieved by scheduling preemptively by LP-basedrandom α-points. Chekuri et al. [6] prove that the optimal nonpreemptive value is atmost e/(e− 1) times the optimal preemptive value; our example in section 3.6 showsthat this bound is tight.

Dyer and Wolsey [9] also propose a (nonpreemptive) time-indexed relaxationwhich is stronger than the preemptive version studied here. This relaxation involvesvariables for each job and each time representing whether this job is being completed(rather than simply processed) at that time. This relaxation is at least as strong asthe preemptive version, but its worst-case ratio is not known to be strictly better.

For randomized on-line algorithms, there is also a gap between the known upperand lower bound on the competitive ratios that are given at the end of section 3.5.For deterministic on-line algorithms, the 2-competitve algorithm of Anderson andPotts [2] is optimal.

Acknowledgment. The authors are grateful to an anonymous referee whosecomments helped to improve the presentation of this paper.

REFERENCES

[1] F. Afrati, E. Bampis, C. Chekuri, D. Karger, C. Kenyon, S. Khanna, I. Milis,M. Queyranne, M. Skutella, C. Stein, and M. Sviridenko, Approximation schemesfor minimizing average weighted completion time with release dates, in Proceedings of the40th Annual IEEE Symposium on Foundations of Computer Science, New York, NY, 1999,pp. 32–43.

[2] E. J. Anderson and C. N. Potts, On-line scheduling of a single machine to minimize totalweighted completion time, in Proceedings of the 13th Annual ACM–SIAM Symposium onDiscrete Algorithms, San Francisco, CA, 2002, pp. 548–557.

[3] E. Balas, On the facial structure of scheduling polyhedra, Math. Programming Stud., 24 (1985),pp. 179–218.

[4] H. Belouadah, M. E. Posner, and C. N. Potts, Scheduling with release dates on a singlemachine to minimize total weighted completion time, Discrete Appl. Math., 36 (1992),pp. 213–231.

[5] S. Chakrabarti, C. Phillips, A. S. Schulz, D. B. Shmoys, C. Stein, and J. Wein, Improvedscheduling algorithms for minsum criteria, in Automata, Languages and Programming,Lecture Notes in Comput. Sci. 1099, F. Meyer auf der Heide and B. Monien, eds., Springer,Berlin, 1996, pp. 646–657.

[6] C. Chekuri, R. Motwani, B. Natarajan, and C. Stein, Approximation techniques for aver-age completion time scheduling, SIAM J. Comput., 31 (2001), pp. 146–166.

[7] C.-F. Chou, M. Queyranne, and D. Simchi-Levi, The asymptotic performance ratio of an on-line algorithm for uniform parallel machine scheduling with release dates, in Integer Pro-gramming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 2081, K. Aardaland B. Gerards, eds., Springer, Berlin, 2001, pp. 45–59.

[8] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms,2nd ed., MIT Press, Cambridge, MA, 2001.

[9] M. E. Dyer and L. A. Wolsey, Formulating the single machine sequencing problem withrelease dates as a mixed integer program, Discrete Appl. Math., 26 (1990), pp. 255–270.

[10] M. X. Goemans, A supermodular relaxation for scheduling with release dates, in Integer Pro-gramming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 1084, W. H.


Cunningham, S. T. McCormick, and M. Queyranne, eds., Springer, Berlin, 1996, pp. 288–300.

[11] M. X. Goemans, Improved approximation algorithms for scheduling with release dates, in Pro-ceedings of the 8th Annual ACM–SIAM Symposium on Discrete Algorithms, New Orleans,LA, 1997, ACM, New York, 1997, pp. 591–598.

[12] R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, Optimizationand approximation in deterministic sequencing and scheduling: A survey, Ann. DiscreteMath., 5 (1979), pp. 287–326.

[13] L. A. Hall, A. S. Schulz, D. B. Shmoys, and J. Wein, Scheduling to minimize averagecompletion time: Off-line and on-line approximation algorithms, Math. Oper. Res., 22(1997), pp. 513–544.

[14] L. A. Hall, D. B. Shmoys, and J. Wein, Scheduling to minimize average completion time:Off-line and on-line algorithms, in Proceedings of the 7th Annual ACM–SIAM Symposiumon Discrete Algorithms, Atlanta, GA, 1996, ACM, New York, 1996, pp. 142–151.

[15] J. A. Hoogeveen and A. P. A. Vestjens, Optimal on-line algorithms for single-machinescheduling, in Integer Programming and Combinatorial Optimization, Lecture Notes inComput. Sci. 1084, W. H. Cunningham, S. T. McCormick, and M. Queyranne, eds.,Springer, Berlin, 1996, pp. 404–414.

[16] J. Labetoulle, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, Preemptivescheduling of uniform machines subject to release dates, in Progress in CombinatorialOptimization, W. R. Pulleyblank, ed., Academic Press, New York, 1984, pp. 245–261.

[17] J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker, Complexity of machine schedulingproblems, Ann. Discrete Math., 1 (1977), pp. 343–362.

[18] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, Cam-bridge, UK, 1995.

[19] A. Munier, M. Queyranne, and A. S. Schulz, Approximation bounds for a general class ofprecedence constrained parallel machine scheduling problems, in Integer Programming andCombinatorial Optimization, Lecture Notes in Comput. Sci. 1412, R. E. Bixby, E. A. Boyd,and R. Z. Rıos-Mercado, eds., Springer, Berlin, 1998, pp. 367–382, SIAM J. Comput., toappear.

[20] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, An analysis of approximations formaximizing submodular set functions. I, Math. Programming, 14 (1978), pp. 265–294.

[21] C. Phillips, C. Stein, and J. Wein, Minimizing average completion time in the presence ofrelease dates, Math. Programming, 82 (1998), pp. 199–223.

[22] M. Queyranne, Structure of a simple scheduling polyhedron, Math. Programming, 58 (1993),pp. 263–285.

[23] M. Queyranne and A. S. Schulz, Polyhedral Approaches to Machine Scheduling, Preprint408, Department of Mathematics, Technische Universitat Berlin, Germany, 1994.

[24] M. Queyranne and A. S. Schulz, Scheduling unit jobs with compatible release dates onparallel machines with nonstationary speeds, in Integer Programming and CombinatorialOptimization, Lecture Notes in Comput. Sci. 920, Springer, Berlin, 1995, pp. 307–320.

[25] M. W. P. Savelsbergh, R. N. Uma, and J. M. Wein, An experimental study of LP-basedapproximation algorithms for scheduling problems, in Proceedings of the 9th Annual ACM–SIAM Symposium on Discrete Algorithms, San Francisco, CA, 1998, ACM, New York,1998, pp. 453–462.

[26] A. S. Schulz, Scheduling to minimize total weighted completion time: Performance guaran-tees of LP-based heuristics and lower bounds, in Integer Programming and CombinatorialOptimization, Lecture Notes in Comput. Sci. 1084, W. H. Cunningham, S. T. McCormick,and M. Queyranne, eds., Springer, Berlin, 1996, pp. 301–315.

[27] A. S. Schulz and M. Skutella, Random-based scheduling: New approximations and LP lowerbounds, in Randomization and Approximation Techniques in Computer Science, LectureNotes in Comput. Sci. 1269, J. Rolim, ed., Springer, Berlin, 1997, pp. 119–133.

[28] A. S. Schulz and M. Skutella, Scheduling-LPs bear probabilities: Randomized approxima-tions for min-sum criteria, in Algorithms—ESA ’97, Lecture Notes in Comput. Sci. 1284,Springer, Berlin, 1997, pp. 416–429, SIAM J. Discrete Math., to appear under the title“Scheduling unrelated machines by randomized rounding.”

[29] A. S. Schulz and M. Skutella, The power of α-points in preemptive single machine schedul-ing, J. Sched., 5 (2002), to appear.

[30] J. Sgall, On-line scheduling—a survey, in Online Algorithms: The State of the Art, LectureNotes in Comput. Sci. 1441, A. Fiat and G. J. Woeginger, eds., Springer, Berlin, 1998,pp. 196–231.

[31] L. S. Shapley, Cores of convex games, Internat. J. Game Theory, 1 (1971), pp. 11–26.


[32] M. Skutella, Approximation and Randomization in Scheduling, Ph.D. thesis, Technische Uni-versitat Berlin, Germany, 1998.

[33] M. Skutella, Convex quadratic and semidefinite programming relaxations in scheduling, J.ACM, 48 (2001), pp. 206–242.

[34] M. Skutella and G. J. Woeginger, A PTAS for minimizing the total weighted completiontime on identical parallel machines, Math. Oper. Res., 25 (2000), pp. 63–75.

[35] W. E. Smith, Various optimizers for single-stage production, Naval Res. Logist. Quart., 3(1956), pp. 59–66.

[36] L. Stougie, private communication in [15], 1995.[37] L. Stougie and A. P. A. Vestjens, Randomized algorithms for on-line scheduling problems:

How low can’t you go?, Oper. Res. Lett., to appear.[38] R. N. Uma and J. M. Wein, On the relationship between combinatorial and LP-based ap-

proaches to NP-hard scheduling problems, in Integer Programming and Combinatorial Op-timization, Lecture Notes in Comput. Sci. 1412, R. E. Bixby, E. A. Boyd, and R. Z.Rıos-Mercado, eds., Springer, Berlin, 1998, pp. 394–408.

[39] A. P. A. Vestjens, On-Line Machine Scheduling, Ph.D. thesis, Eindhoven University of Tech-nology, The Netherlands, 1997.

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